考研数学基本公式

等价无穷小

函数 无穷小 函数 无穷小
\(\sin x\) $x$ \(1-\cos x\) $\dfrac{1}{2}x^{2}$
\(\tan x\) $x$ $x-\ln \left( 1+x\right) $ $\dfrac{1}{2}x^{2}$
\(\arcsin x\) $x$ \(x-\sin x\) $\dfrac{1}{6}x^{3}$
\(\arctan x\) $x$ \(\arcsin x-x\) $\dfrac{1}{6}x^{3}$
$\ln \left( 1+x\right) $ $x$ \(\tan x-x\) $\dfrac{1}{3}x^{3}$
\(e^x-1\) $x$ \(x-\arctan x\) $\dfrac{1}{3}x^{3}$
\(\sqrt{1+x}-\sqrt{1-x}\) $x$ \(\sqrt[n] {1+x}-1\) $\dfrac{1}{n}x$
\(a^x-1\) $x\ln a$ \(\left( 1+x\right) ^{\alpha }-1\) $\alpha x$

求导公式

函数 导数 函数 导数
\(\left( C\right)^{’}\) \(0\) \(\left( x^{\alpha } \right) ^{’}\) \(\alpha x^{\alpha -1}\)
\((a^{x})^{’}\) \(a^{x}\ln a\) \((e^{x})^{’}\) \(e^{x}\)
$\left( \log _{a}x\right) ^{’} $ $\dfrac{1}{x\ln a} $ \(\left( \ln \left\vert x\right\vert \right) ^{’}\) \(\dfrac{1}{x}\)
\((\sin x)^{’}\) \(\cos x\) \((\cos x)^{’}\) \(-\sin x\)
$(\tan x)^{’} $ \(\sec ^{2}x\) $(\cot x)^{’} $ \(-\csc ^{2}x\)
$(\sec x)^{’} $ \(\sec x\tan x\) $(\csc x)^{’} $ \(-\csc x\cot x\)
$(\arcsin x)^{’} $ $\dfrac{1}{\sqrt{1-x^{2}}} $ $(\arccos x)^{’} $ $-\dfrac{1}{\sqrt{1-x^{2}}} $
$(\arctan x)^{’} $ $\dfrac{1}{1+x^{2}} $ \(\left( arccotx\right) ^{’}\) $-\dfrac{1}{1+x^{2}} $

常用泰勒展开

泰勒展开\(x\rightarrow 0\)
\(e^{x}=1+x+\dfrac{x^{2}}{2!}+\dfrac{x^{3}}{3!}+\ldots +\dfrac{x^{n}}{n!}+o\left( x^{n}\right)\)
\(\sin x=x-\dfrac{x^{3}}{3!}+\dfrac{x^{5}}{5!}+\ldots +\left( -1\right) ^{n-1}\dfrac{x^{2n-1}}{\left( 2n-1\right) !}+o\left( x^{2n-1}\right)\)
\(\cos x=1-\dfrac{x^{2}}{2!}+\dfrac{x^{4}}{4!}+\ldots +\left( -1\right) ^{n}\dfrac{x^{2n}}{\left( 2n\right) !}+o\left( x^{2n}\right)\)
\(\ln \left( 1+x\right) =x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}+\ldots +\left( -1\right) ^{n-1}\dfrac{x^{n}}{n}+o\left( x^{n}\right)\)
\(\left( 1+x\right) ^{\alpha }=1+\alpha x+\dfrac{2\left( \alpha -1\right) }{2!}x^{2}+\ldots +\dfrac{\alpha \left( \alpha -1\right) \ldots \left( \alpha -n+1\right) }{n!}x^{n}+o\left( x^{n}\right)\)

积分公式

积分公式 积分公式
\(\int 0 dx=C\) \(\int 1 dx=\int dx=x+C\)
\(\int x^{\alpha }dx=\dfrac{1}{\alpha +1}x^{\alpha +1}+C( \alpha \neq -1)\) $\int \dfrac{1}{x}dx=\ln \left\vert x\right\vert +C $
\(\int a^{x}dx=\dfrac{a^{x}}{\ln a}+C (a>0,a\neq 1)\) \(\int e^{x}dx=e^{x}+C\)
\(\int \sin xdx=-\cos x+C\) \(\int \cos xdx=\sin x+C\)
\(\int \tan xdx=-\ln \vert \cos x \vert +C\) \(\int \cot xdx=\ln \vert \sin x \vert +C\)
\(\int \sec xdx=\ln \vert \sec x+\tan x \vert +C\) \(\int \csc xdx=\ln \vert \csc x-\cot x \vert +C\)
\(\int \sec ^{2}xdx=\tan x+C\) \(\int \csc ^{2}xdx=-\cot x+C\)
\(\int \dfrac{1}{a^{2}+x^{2}}dx=\dfrac{1}{a}\arctan \dfrac{x}{a}+C\) \(\int \dfrac{1}{a^{2}-x^{2}}dx=\dfrac{1}{2a}\ln \left \vert \dfrac{a+x}{a-x}\right \vert +C\)
\(\int \dfrac{1}{\sqrt{a^{2}-x^{2}}}dx=\arcsin \dfrac{x}{a}+C\) \(\int \dfrac{1}{\sqrt{x^{2}\pm a^{2}}}dx=\ln \left \vert x+\sqrt{x^{2}\pm a^{2}}\right \vert +C\)

数列极限

  • 数列极限收敛一定有界，有界不一定收敛
  • 数列极限如果存在，则极限值与前有限项无关
  • 若一个数列收敛于\(a\)，则其任一子数列也收敛于\(a\).
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常见反常积分的敛散性

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柯西中值定理

定理设$f(x)$、$g(x)$在闭区间$[a,b]$上连续,在开区间$(a,b)$内可导，且
$g^{\prime}(x)\neq0,,x\in(a,b)$ ,则至少 存在一点$ \zeta$∈(a,b),使${\frac{f(b)-f(a)}{g(b)-g(a)}}={\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}} $

【注】 柯西中值定理是拉格朗日中值定理在两个函数情形下的推广.

泰勒公式

$f(x)=f(x_{0})+f^{\prime}(x_{0})(x-x_{0})+{\frac{f^{\prime\prime}(x_{0})}{2!}}(x-x_{0})^{2}+\cdots+{\frac{f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}+o((x-x_{0})^{n}).$

更新信息

日期 版本 更新内容
2024-10-02 1.1 常见反常积分的敛散性
2024-10-01 1.1 新功能
修复问题
2024-10-01 1.0 转载自 今天你学了吗
优化性能